[1208] | 1 | \section[Ionization]{Ionization} |
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| 2 | |
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| 3 | \subsection{Method} |
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| 4 | |
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| 5 | The class $G4hIonisation$ provides the continuous energy loss due to |
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| 6 | ionization and simulates the 'discrete' part of the ionization, that is, |
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| 7 | delta rays produced by charged hadrons. The class $G4ionIonisation$ is |
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| 8 | intended for the simulation of energy loss by ions and the approach described |
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| 9 | in Section \ref{en_loss} is used. The value of the maximum energy |
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| 10 | transferable to a free electron $T_{max}$ is given by the following relation: |
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| 11 | \begin{equation} |
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| 12 | \label{hion.c} |
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| 13 | T_{max} =\frac{2mc^2(\gamma^2 -1)}{1+2\gamma (m/M)+(m/M)^2 }, |
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| 14 | \end{equation} |
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| 15 | where $m$ is the electron mass and $M$ is the mass of the incident particle. |
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| 16 | The method of calculation of the continuous energy loss and the total |
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| 17 | cross-section are explained below. |
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| 18 | |
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| 19 | \subsection{Continuous Energy Loss} |
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| 20 | |
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| 21 | The integration of \ref{comion.a} leads to the Bethe-Bloch restricted energy |
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| 22 | loss formula \cite{hion.pdg} : |
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| 23 | \begin{equation} |
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| 24 | \label{hion.d} |
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| 25 | \left. \frac{dE}{dx} \right]_{T < T_{cut}} = |
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| 26 | 2 \pi r_e^2 mc^2 n_{el} \frac{(z_p)^2}{\beta^2} |
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| 27 | \left [\ln \left(\frac{2mc^2 \beta^2 \gamma^2 T_{up}} {I^2} \right) |
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| 28 | - \beta^2 \left( 1 + \frac{T_{up}}{T_{max}} \right) |
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| 29 | - \delta - \frac{2C_e}{Z} \right ] |
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| 30 | \end{equation} |
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| 31 | where |
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| 32 | \[ |
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| 33 | \begin{array}{ll} |
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| 34 | r_e & \mbox{classical electron radius:} |
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| 35 | \quad e^2/(4 \pi \epsilon_0 mc^2 ) \\ |
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| 36 | mc^2 & \mbox{mass-energy of the electron} \\ |
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| 37 | n_{el} & \mbox{electrons density in the material} \\ |
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| 38 | I & \mbox{mean excitation energy in the material}\\ |
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| 39 | \gamma & \mbox{$E/mc^2$} \\ |
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| 40 | \beta^2 & 1-(1/\gamma^2) \\ |
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| 41 | T_{up} & \min(T_{cut},T_{max}) \\ |
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| 42 | \delta & \mbox{density effect function} \\ |
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| 43 | C_e & \mbox{shell correction function} |
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| 44 | \end{array} |
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| 45 | \] |
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| 46 | In a single element the electron density is |
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| 47 | $$ n_{el} = Z \: n_{at} = Z \: \frac{\mathcal{N}_{av} \rho}{A} $$ |
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| 48 | ($\mathcal{N}_{av}$: Avogadro number, $\rho$: density of the material, |
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| 49 | $A$: mass of a mole). In a compound material |
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| 50 | $$ |
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| 51 | n_{el} = \sum_i Z_i \: n_{ati} |
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| 52 | = \sum_i Z_i \: \frac{\mathcal{N}_{av} w_i \rho}{A_i} . |
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| 53 | $$ |
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| 54 | $w_i$ is the proportion by mass of the $i^{th}$ element, with molar mass $A_i$. |
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| 55 | |
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| 56 | |
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| 57 | The mean excitation energy $I$ for all elements is tabulated according to |
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| 58 | the ICRU recommended values \cite{hion.icru1}. |
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| 59 | |
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| 60 | \subsubsection{Density Correction} |
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| 61 | |
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| 62 | $\delta$ is a correction term which takes into account the reduction in energy |
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| 63 | loss due to the so-called {\it density effect}. This becomes important at |
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| 64 | high energies because media have a tendency to become polarized as the |
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| 65 | incident particle velocity increases. As a consequence, the atoms in a |
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| 66 | medium can no longer be considered as isolated. To correct for this effect |
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| 67 | the formulation of Sternheimer~\cite{hion.sternheimer} is used: |
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| 68 | \input{electromagnetic/utils/densityeffect} |
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| 69 | |
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| 70 | \subsubsection{Shell Correction} |
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| 71 | |
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| 72 | $2C_e/Z$ is the so-called {\it shell correction term} which accounts for the |
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| 73 | fact that, at low energies for light elements and at all energies for heavy |
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| 74 | ones, the probability of collision with the electrons of the inner atomic |
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| 75 | shells (K, L, etc.) is negligible. The semi-empirical formula used |
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| 76 | in {\sc Geant4}, applicable to all materials, is due to |
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| 77 | Barkas \cite{hion.barkas}: |
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| 78 | \begin{equation} |
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| 79 | \label{hion.dd} |
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| 80 | C_e(I, \beta\gamma) = \frac{a(I)}{(\beta\gamma)^2} |
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| 81 | +\frac{b(I)}{(\beta\gamma)^4} |
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| 82 | +\frac{c(I)}{(\beta\gamma)^6} . |
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| 83 | \end{equation} |
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| 84 | The functions a(I), b(I) and c(I) can be found in the source code. This |
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| 85 | formula breaks down at low energies, and is valid only when |
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| 86 | $\beta\gamma > 0.13$ ($T > 7.9$ MeV for a proton). For $\beta\gamma \leq |
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| 87 | 0.13$ the shell correction term is calculated as: |
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| 88 | \begin{equation} |
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| 89 | \label{hion.ddd} |
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| 90 | \left . C_{e}(I,\beta\gamma) \rule{0mm}{5mm} \right |_{\beta\gamma \leq 0.13} |
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| 91 | = C_{e}(I,\beta\gamma=0.13)\frac{\ln(T/T_{2l})}{\ln(7.9 \: \rm MeV/T_{2l})}, |
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| 92 | \end{equation} |
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| 93 | i.e. the correction is switched off logarithmically from $T=7.9$ MeV |
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| 94 | to $T=T_{2l}=2$ MeV. |
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| 95 | |
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| 96 | \subsubsection{Parameterization} |
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| 97 | |
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| 98 | The mean energy loss can be described by the Bethe-Bloch formula |
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| 99 | (\ref{muion1}) only if the projectile velocity is larger than that of the |
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| 100 | orbital electrons. In the low-energy region this is not the case, and the |
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| 101 | parameterization from the ICRU'49 report \cite{hion.ICRU49} is used in the |
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| 102 | $G4BraggModel$ class. The Bethe-Bloch model is applied for higher kinetic |
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| 103 | energies of incident particles |
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| 104 | \begin{equation} |
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| 105 | \label{muion.lowen1} |
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| 106 | T > 2 * M/M_{proton} MeV, |
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| 107 | \end{equation} |
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| 108 | where $M$ is the particle mass. The details of the low energy |
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| 109 | parameterization are described in Section \ref{le_had_ion}. |
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| 110 | |
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| 111 | |
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| 112 | \subsection{Total Cross Section per Atom and Mean Free Path} |
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| 113 | |
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| 114 | For $T \gg I $ the differential cross section can be written as |
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| 115 | \begin{equation} |
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| 116 | \label{hion.i} |
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| 117 | \frac{d\sigma }{dT} = 2\pi r_e^2 mc^2 Z \frac{z_p^2}{\beta^2} \frac{1}{T^2} |
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| 118 | \left[ 1 - \beta^2 \frac{T}{T_{max}} + \frac{T^2}{2E^2} \right] |
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| 119 | \end{equation} |
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| 120 | \cite{hion.pdg}. In {\sc Geant4} $T_{cut} \geq 1$ keV. Integrating from |
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| 121 | $T_{cut}$ to $T_{max}$ gives the total cross section per atom : |
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| 122 | \begin{eqnarray} |
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| 123 | \label{hion.j} |
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| 124 | \sigma (Z,E,T_{cut}) & = & \frac {2\pi r_e^2 Z z_p^2}{\beta^2} mc^2 \times |
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| 125 | \\ & & \left[ \left( \frac{1}{T_{cut}} - \frac{1}{T_{max}} \right) |
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| 126 | - \frac{\beta^2}{T_{max}} \ln \frac{T_{max}}{T_{cut}} |
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| 127 | + \frac{T_{max} - T_{cut}}{2E^2} |
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| 128 | \right] \nonumber |
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| 129 | \end{eqnarray} |
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| 130 | The last term is for spin $1/2$ only. In a given material the mean free path |
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| 131 | is: |
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| 132 | \begin{equation} |
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| 133 | \begin{array}{lll} |
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| 134 | \lambda = (n_{at} \cdot \sigma)^{-1} & or & |
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| 135 | \lambda = \left( \sum_i n_{ati} \cdot \sigma_i \right)^{-1} |
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| 136 | \end{array} |
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| 137 | \end{equation} |
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| 138 | The mean free path is tabulated during initialization as a function of the |
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| 139 | material and of the energy for all kinds of charged particles. |
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| 140 | |
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| 141 | \subsection{Simulating Delta-ray Production} |
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| 142 | |
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| 143 | A short overview of the sampling method is given in Chapter \ref{secmessel}. |
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| 144 | Apart from the normalization, the cross section \ref{hion.i} can be |
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| 145 | factorized : |
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| 146 | \begin{eqnarray} |
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| 147 | \frac{d\sigma}{dT}=f(T) g(T) &with& T \in [T_{cut}, \ T_{max}] |
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| 148 | \end{eqnarray} |
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| 149 | where |
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| 150 | \begin{eqnarray} |
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| 151 | f(T) &=& \left(\frac{1}{T_{cut}} - \frac{1}{T_{max}} \right) \frac{1}{T^2} \\ |
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| 152 | g(T) &=& 1 - \beta^2 \frac{T}{T_{max}} + \frac{T^2}{2E^2} . |
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| 153 | \end{eqnarray} |
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| 154 | The last term in $g(T)$ is for spin $1/2$ only. The energy $T$ is chosen by |
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| 155 | \begin{enumerate} |
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| 156 | \item sampling $T$ from $f(T)$ |
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| 157 | \item calculating the rejection function $g(T)$ and accepting the sampled |
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| 158 | $T$ with a probability of $g(T)$. |
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| 159 | \end{enumerate} |
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| 160 | After the successful sampling of the energy, the direction |
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| 161 | of the scattered electron is generated with respect to the direction of the |
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| 162 | incident particle. The azimuthal angle $\phi$ is generated isotropically. |
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| 163 | The polar angle $\theta$ is calculated from energy-momentum conservation. |
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| 164 | This information is used to calculate the energy and momentum of both |
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| 165 | scattered particles and to transform them into the {\em global} coordinate |
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| 166 | system. |
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| 167 | |
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| 168 | \subsubsection{Ion Effective Charge} |
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| 169 | |
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| 170 | As ions penetrate matter they exchange electrons with the medium. In the |
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| 171 | implementation of $G4ionIonisation$ the effective charge approach is |
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| 172 | used \cite{hion.Ziegler85}. |
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| 173 | A state of equilibrium between the ion and the medium is assumed, so that |
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| 174 | the ion's effective charge can be calculated as a function of its kinetic |
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| 175 | energy in a given material. This is done according to the approximation |
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| 176 | described in Section \ref{le_had_ion}. Before and after each step the dynamic |
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| 177 | charge of the ion is recalculated and saved in $G4DynamicParticle$, where |
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| 178 | it can be used not only for energy loss calculations but also for the |
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| 179 | sampling of transportation in an electromagnetic field. |
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| 180 | |
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| 181 | |
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| 182 | \subsection{Status of this document} |
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| 183 | 09.10.98 created by L. Urb\'an. \\ |
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| 184 | 14.12.01 revised by M.Maire \\ |
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| 185 | 29.11.02 re-worded by D.H. Wright \\ |
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| 186 | 01.12.03 revised by V. Ivanchenko \\ |
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| 187 | |
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| 188 | \begin{latexonly} |
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| 189 | |
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| 190 | \begin{thebibliography}{99} |
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| 191 | |
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| 192 | \bibitem{hion.pdg} |
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| 193 | Particle Data Group. Rev. of Particle Properties. |
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| 194 | Eur. Phys. J. C15. (2000) 1. http://pdg.lbl.gov |
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| 195 | \bibitem{hion.icru1} |
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| 196 | ICRU Report No. 37 (1984) |
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| 197 | \bibitem{hion.sternheimer} |
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| 198 | R.M.Sternheimer. Phys.Rev. B3 (1971) 3681. |
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| 199 | \bibitem{hion.barkas} |
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| 200 | W. H. Barkas. Technical Report 10292,UCRL, August 1962. |
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| 201 | \bibitem{hion.ICRU49}ICRU (A.~Allisy et al), |
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| 202 | Stopping Powers and Ranges for Protons and Alpha |
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| 203 | Particles, |
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| 204 | ICRU Report 49, 1993. |
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| 205 | \bibitem{hion.Ziegler85}J.F.~Ziegler, J.P.~Biersack, U |
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| 206 | .~Littmark, The Stopping |
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| 207 | and Ranges of Ions in Solids. Vol.1, Pergamon Press, 1985. |
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| 208 | \end{thebibliography} |
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| 209 | |
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| 210 | \end{latexonly} |
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| 211 | |
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| 212 | \begin{htmlonly} |
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| 213 | |
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| 214 | \subsection{Bibliography} |
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| 215 | |
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| 216 | \begin{enumerate} |
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| 217 | \item Particle Data Group. Rev. of Particle Properties. |
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| 218 | Eur. Phys. J. C15. (2000) 1. http://pdg.lbl.gov |
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| 219 | \item ICRU Report No. 37 (1984) |
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| 220 | \item R.M.Sternheimer. Phys.Rev. B3 (1971) 3681. |
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| 221 | \item W.H. Barkas. Technical Report 10292,UCRL, August 1962. |
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| 222 | \item ICRU (A.~Allisy et al), |
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| 223 | Stopping Powers and Ranges for Protons and Alpha Particles, |
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| 224 | ICRU Report 49, 1993. |
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| 225 | \item J.F.~Ziegler, J.P.~Biersack, U.~Littmark, The Stopping |
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| 226 | and Ranges of Ions in Solids. Vol.1, Pergamon Press, 1985. |
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| 227 | \end{enumerate} |
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| 228 | |
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| 229 | \end{htmlonly} |
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| 230 | |
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| 231 | |
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